The Fundamental Crossed Module of the Complement of a Knotted Surface

TL;DR

This paper develops a homotopy-invariant algebraic framework for analyzing the topology of knotted surface complements in four-dimensional space, introducing an algorithm to compute a fundamental algebraic structure from geometric data.

Contribution

It introduces a new method to compute the algebraic 2-type of knotted surface complements using crossed modules and provides an algorithm based on handle decompositions.

Findings

The invariant $I_G$ is non-trivial and computable for knotted surfaces.

The method relates geometric handle decompositions to algebraic invariants.

The approach generalizes to finite crossed modules and homotopy types.

Abstract

We prove that if $M$ is a CW-complex and $M^1$ is its 1-skeleton then the crossed module $\Pi_2(M,M^1)$ depends only on the homotopy type of $M$ as a space, up to free products, in the category of crossed modules, with $\Pi_2(D^2,S^1)$. From this it follows that, if $G$ is a finite crossed module and $M$ is finite, then the number of crossed module morphisms $\Pi_2(M,M^1) \to G$ can be re-scaled to a homotopy invariant $I_G(M)$, depending only on the homotopy 2-type of $M$. We describe an algorithm for calculating $\pi_2(M,M^{(1)})$ as a crossed module over $\pi_1(M^{(1)})$, in the case when $M$ is the complement of a knotted surface $\Sigma$ in $S^4$ and $M^{(1)}$ is the handlebody made from the 0- and 1-handles of a handle decomposition of $M$. Here $\Sigma$ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2-type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant $I_G$ yields a non-trivial invariant of knotted surfaces in $S^4$ with good properties with regards to explicit calculations.